Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas

We study theoretically transport of a one-dimensional single-channel interacting electron gas through barriers or constrictions. We find that electrons with repulsive interactions, incident upon a single barrier, are completely reflected at zero temperature. At finite temperature (T), the conductance is shown to vanish as a power of T, and at zero temperature, power-law current-voltage characteristics are predicted. For attractive interactions, we predict perfect transmission at zero temperature, with similar power-law corrections. We also study resonant tunneling through a double-barrier structure and related effects associated with the Coulomb blockade. Resonant peaks in the transmission are possible, provided the interactions are not too strongly repulsive. However, in contrast to resonant tunneling in a noninteracting electron gas, we find that in the presence of interactions the width of the resonance vanishes, as a power of temperature, in the zero-temperature limit. Moreover, the resonance line shapes are shown to be described by a universal scaling function, which has power law, but non-Lorentzian tails. For a particular choice of interaction strengths, we present an exact solution of our model, which verifies the scaling assumptions and provides an explicit expression for the scaling function. We also consider the role played by the electron-spin degree of freedom in modifying the trasnsmission through barriers. With spin, there are four possible phases corresponding to perfect transmission or perfect reflection of charge and spin. We present phase diagrams for these different behaviors and analyze the nontrivial transitions between them. At these transitions we find that the conductance or transmission is universal---depending only on the dimensionless conductance of the leads and not on the details of the barriers. In the case of resonant tunneling with spin, we discuss the ``Kondo'' resonance, which occurs when there is a spin degeneracy for electrons between the two barriers. Many of the predictions should be directly testable in gated GaAs wires.